distributions

[Nematrian website page: ProbabilityDistributionsIntro, © Nematrian 2020]

The Nematrian website contains information and analytics on a wide range of probability distributions, including:

Discrete () distributions

- Bernoulli, see also - Binomial - Geometric, see also negative binomial distribution - Hypergeometric - Logarithmic - Negative binomial - Poisson - Uniform (discrete)

Continuous (univariate) distributions

- Beta - Beta prime - Burr - Cauchy - Chi-squared - Dagum - Degenerate - - Exponential - F - Fatigue, also known as the Birnbaum-Saunders distribution - Fréchet, see also generalised extreme value (GEV) distribution - Gamma - Generalised extreme value (GEV) - Generalised gamma - Generalised inverse Gaussian - Generalised Pareto (GDP) - Gumbel, see also generalised extreme value (GEV) distribution - Hyperbolic secant - Inverse gamma - Inverse Gaussian - Johnson SU - Kumaraswamy - Laplace - Lévy - Logistic - Log-logistic - Lognormal - Nakagami - Non-central chi-squared - Non-central t - Normal - Pareto - Power function - Rayleigh - Reciprocal - Rice - Student’s t - Triangular - Uniform - Weibull, see also generalised extreme value (GEV) distribution

Continuous multivariate distributions

- Inverse Wishart

Copulas (a copula is a special type of continuous multivariate distribution)

- Clayton - Comonotonicity - Countermonotonicity (only valid for 푛 = 2, where 푛 is the dimension of the input) - Frank - Generalised Clayton - Gumbel - Gaussian - Independence - t

The Nematrian website functions for fitting univariate distributions, creating random variates and calculating moments etc. now cover most of the above probability distributions, see ProbabilityDistributionsFunctions. In many cases these functions can cater for the traditional (textbook) forms of these distributions and variants that include additional shift and scale .

location (i.e. shift) and scale variants [ProbabilityDistributionsIntro2]

The location and scale of any can be adjusted by using the (linear) transform 푌 = 푔 + ℎ푋 where 푔 and ℎ are constants (푔 adjusts the location, i.e. shifts the distribution, whilst ℎ adjusts its scale). This leaves the skew and (excess ) unaltered but alters the and as 퐸(푌) = 푔 + ℎ퐸(푋) and 푣푎푟(푌) = ℎ2푣푎푟(푋).

In some cases the typical distributional specification already includes such components. For example, the 푁(휇, 𝜎2) is the location and scale adjusted version of the unit normal distribution 푁(0,1).

In other cases the standard distributional specification does not include such adjustments. For example, the (standard) Student’s t distribution depends one just one , its degrees of freedom. The probability distribution orientated Nematrian web functions recognise location and/or scale adjusted variants of wide range of standard probability distributions.

Discrete (univariate) distributions

The [BernoulliDistribution]

A is an that has one of two possible outcomes, ‘success’ with probability 푝 and ‘failure’ with probability 1 − 푝. The Bernoulli distribution is a probability distribution that takes the value of 1 if such a trial is a ‘success’ and 0 if it is a ‘failure’.

The Bernoulli distribution is a special case of the binomial distribution, 퐵(푛, 푝), with 푛 = 1. For characteristics of the Bernoulli distribution (e.g. mean, etc.), please refer to the corresponding characteristics for the binomial distribution.

For other probability distributions see here.

The binomial distribution [BinomialDistribution]

The binomial distribution 퐵(푛, 푝) is the discrete probability distribution applicable to the number of successes in a sequence of 푛 independent yes/no each of which has a success probability of 푝. Each individual success/failure experiment is called a Bernoulli trial, so if 푛 = 1 then the binomial distribution is a Bernoulli distribution.

It has the following characteristics:

Distribution name Binomial distribution Common notation 푋~퐵(푛, 푝) Parameters 푛 = number of (independent) trials, positive integer 푝 = probability of success in each trial, 0 ≤ 푝 ≤ 1 푥 ∈ {0,1, … , 푛} = 푛푢푚푏푒푟 표푓 푠푢푐푐푒푠푠푒푠 Probability mass function 푛 푛! 푓(푥) = ( ) 푝푥(1 − 푝)푛−푥 = 푝푥(1 − 푝)푛−푥 푥 (푛 − 푥)! 푥! Cumulative distribution 푥 푛 푗 푛−푗 function 퐹(푥) = ∑ (푗 ) 푝 (1 − 푝) = 퐼1−푝(푛 − 푥, 푥 + 1) 푗=0 Mean 푛푝 Variance 푛푝(1 − 푝) 1 − 2푝

√푛푝(1 − 푝) (Excess) kurtosis 1 − 6푝(1 − 푝)

푛푝(1 − 푝) 푛 Characteristic function (1 − 푝 + 푝푒푖푡) Other comments The Bernoulli distribution is 퐵(1, 푝) and corresponds to the likelihood of success of a single experiment. Its probability mass function and cumulative distribution function are: 1 − 푝, 푥 = 0 푓(푥) = 퐹(푥) = { 푝, 푥 = 1

The Bernoulli distribution with 푝 = 1⁄2, i.e. 퐵(1, 1⁄2), has the minimum possible excess kurtosis, i.e. −2.

The of 퐵(푛, 푝) is 푖푛푡((푛 + 1)푝) if (푛 + 1)푝 is 0 or not an integer and is 푛 if (푛 + 1)푝 = 푛 + 1. If (푛 + 1)푝 ∈ {1,2, … , 푛} then the distribution is bi-modal, with modes (푛 + 1)푝 and (푛 + 1)푝 − 1.

The binomial distribution is often used to model the number of successes in a sample size of 푛 from a population size of 푁. Since such samples are not independent, the resulting distribution is actually a hypergeometric distribution and not a binomial distribution. However if 푁 ≫ 푛 then the binomial distribution becomes a good approximation to the relevant hypergeometric distribution and is thus often used.

푛 In the above ( ) is the binomial coefficient. 푥

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “binomial”. For details of other supported probability distributions see here.

The [GeometricDistribution]

The geometric distribution describes the probability of 푥 successes in a sequence of independent experiments each with likelihood of success of 푝 that arise before there is 1 failure. It is a special case of the negative binomial distribution.

Note: different texts adopt slightly different definitions, e.g. it may be the total number of trials (i.e. 1 more than the above) in which case it may be called the shifted geometric distribution and/or 푝 may denote the probability of failure rather than the probability of success.

The hypergeometric distribution [HypergeometricDistribution]

The hypergeometric distribution describes the probability of 푥 successes in 푛 draws from a finite population size 푁 containing 푚 successes without replacement. This contrasts with the binomial distribution which describes the probability of 푥 successes in 푛 draws with replacement

Distribution name Hypergeometric distribution Common notation 푋~퐻푦푝푒푟푔푒표푚푒푡푟푖푐(푚, 푁, 푛) Parameters 푁 = population size, integral (푁 > 0) 푛 = sample size, integral (0 < 푛 ≤ 푁) 푚 = number of tagged items, integral (0 < 푚 ≤ 푁) Domain max(0, 푛 + 푚 − 푁) ≤ 푥 ≤ max(푛, 푚) , 푥 an integer 푚 푁 − 푚 Probability mass function ( ) ( ) 푥 푓(푥) = 푛 − 푥 푁 ( ) 푛 Cumulative distribution 푚 푁 − 푚 푛 푁 − 푛 푥 ( ) ( ) ( ) ( ) 푗 푛 − 푗 푘 + 1 function 퐹(푥) = (푥) = ∑ = 1 − 푚 − 푘 − 1 푌 푁 푁 ( ) ( ) 푗=0 푗 푚 where

푌 = 3퐹2(1, 푥 + 1 − 푚, 푥 + 1 − 푛; 푥 + 2, 푁 + 푥 + 2 − 푚 − 푛; 1) 푝퐹푞 is the generalised hypergeometric function, i.e. ∞ (푎 ) … (푎 ) 푛 1 푛 푝 푛 푧 푝퐹푞(푎1, … , 푎푝; 푏1, … , 푏푞; 푧) = ∑ (푏 ) … (푏 ) 푛! 푛=0 1 푛 푞 푛 and (푎)푛 involves the rising factorial or Pochhammer notation, i.e. (푎)푛 = 푎(푎 + 1)(푎 + 2) … (푎 + 푛 − 1) and (푎)0 = 1 푛푚 Mean 푁 Variance 푛푚(푁 − 푚)(푁 − 푛)

푁2(푁 − 1) Skewness (푁 − 2푚)(푁 − 1)1⁄2(푁 − 2푛) 1⁄2 (푛푚(푁 − 푚)(푁 − 푛)) (푁 − 2) (Excess) kurtosis 퐴 + 퐵

퐶 where 퐴 = (푁 − 1)푁2(푁(푁 + 1) − 6푚(푁 − 푚) − 6푛(푁 − 푚)) 퐵 = 6푛푚(푁 − 푚)(푁 − 푛)(5푁 − 6) 퐶 = 푛푚(푁 − 푚)(푁 − 푛)(푁 − 2)(푁 − 3) 푁 − 푚 Characteristic function ( ) 푛 퐹 (−푛, −푚; 푛 − 푚 − 푛 + 1; 푒푖푡) 푁 2 1 ( ) 푛

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “hypergeometric”. For details of other supported probability distributions see here.

The [LogarithmicDistribution]

The logarithmic distribution arises from following power series expansion:

푝2 푝3 − log(1 − 푝) = 푝 + + + ⋯ 2 3

푝푥 This that the function 푓(푥) = − , 푥 = 1,2,3, … can naturally be interpreted as a 푥 log(1−푝) ∞ probability mass function since ∑푘=1 푓(푘) = 1.

Distribution name Logarithmic distribution Common notation 푋~퐿표푔(푝) Parameters 푝 = (0 < 푝 < 1) Domain 1 ≤ 푥 < +∞, 푥 an integer Probability mass function 푝푥 푓(푥) = − 푥 log(1 − 푝) Cumulative distribution 푥 푗 1 푝 퐵푝(푥 + 1,0) 퐹(푥) = − ∑ = 1 + function log(1 − 푝) 푗 log(1 − 푝) 푗=1 푝 Mean − (1 − 푝) log(1 − 푝) Variance 푝(푝 + log(1 − 푝)) − = 푉 (1 − 푝)2(log(1 − 푝))2 Skewness 푝 3푝 2푝2 − (1 + 푝 + + ) (1 − 푝)3푉3⁄2 log(1 − 푝) log(1 − 푝) log2(1 − 푝) 푝 (Excess) kurtosis − 퐴 − 3 (1 − 푝)4푉2 log(1 − 푝) where 4푝(1 + 푝) 6푝2 3푝3 퐴 = (1 + 4푝 + 푝2 + + + ) log(1 − 푝) log2(1 − 푝) log3(1 − 푝) Characteristic function log(1 − 푝푒푖푡)

log(1 − 푝) Other comments The logarithmic distribution has a mode of 1. If 푁 is a with Poission distribution and 푋푖, 푖 = 1, … is an infinite sequence of iid random variables each distributed 퐿표푔(푝) then 푌 = 푁 ∑푖=1 푋푖 has a negative binomial distribution showing that the negative binomial distribution is an example of a compound

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “logarithmic”. For details of other supported probability distributions see here.

The negative binomial distribution [NegativeBinomialDistribution]

The negative binomial distribution describes the probability of 푥 successes in a sequence of independent experiments each with likelihood of success of 푝 that arise before there are 푟 failures. In this interpretation 푟 is a positive integer, but the distributional definition can also be extended to real values of 푟 > 0. Note: different texts adopt slightly different definitions, e.g. with support starting at 푥 = 푟 not 푥 = 0 and/or with 푝 denoting probability of failure rather than probability of success.

Distribution name Negative binomial distribution Common notation 푋~푁퐵(푘, 푝) Parameters 푟 = number of failures (푟 > 0) 푝 = probability of success in each experiment (0 < 푝 < 1) Support 푥 ∈ {0,1,2, … } 푥 + 푟 − 1 Probability mass function 푓(푥) = ( ) 푝푥(1 − 푝)푟 푥 If 푟 is non-integral then is: Γ(푥 + 푟) 푓(푥) = 푝푥(1 − 푝)푟 Γ(푟)Γ(푥 + 1) Cumulative distribution 퐹(푥) = 1 − 퐼푝(푥 + 1, 푟) function 푝푟 Mean 1 − 푝 푝푟 Variance (1 − 푝)2 Skewness 1 + 푝

√푝푟 (Excess) kurtosis 6 (1 − 푝)2 + 푟 푝푟 Characteristic function 1 − 푝 푟 ( ) 1 − 푝푒푖푡 Other comments The geometric distribution is the same as the negative binomial distribution with parameter 푟 = 1. Its pdf and cdf are therefore: 푓(푥) = 푝(1 − 푝)푥 퐹(푥) = 1 − (1 − 푝)푥+1

For the special case where 푟 is an integer the negative binomial distribution is also called the Pascal distribution. The Poisson distribution is also a limiting case of the negative binomial: 푟 푃표푖푠푠표푛(휆) = lim 푁퐵 (푟, ) 푟→∞ 휆 + 푟

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “negative binomial”. For details of other supported probability distributions see here.

The Poisson distribution [PoissonDistribution]

The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time if the events occur with a known average rate and independently of the time since the last event.

Distribution name Poisson distribution Common notation 푋~푃표푖푠(휆) Parameters 휆 = event rate (휆 > 0) Support 푥 ∈ {0,1,2, … } Probability mass function 휆푥푒−휆 푓(푥) = 푥! 푥 Cumulative distribution 휆푗 퐹(푥) = 푒−휆 ∑ function 푗! 푗=0 (can also be expressed using the incomplete ) Mean 휆 Variance 휆 Skewness 휆−1⁄2 (Excess) kurtosis 휆−1 푖푡 Characteristic function 푒휆(푒 −1) Other comments The is approximately 푖푛푡(휆 + 1⁄3 − 0.02⁄휆).

The mode is 푖푛푡(휆) if 휆 is not integral. Otherwise the distribution is bi-modal with modes 휆 and 휆 − 1.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “poisson”. For details of other supported probability distributions see here.

The uniform (discrete) distribution [UniformDiscreteDistribution]

The uniform (discrete) distribution involves equally probable outcomes that are spaced uniform intervals apart.

Distribution name Uniform (discrete) distribution Common notation 푋~푈(푎, 푏, ℎ) Parameters 푎 = lower limit 푏 = upper limit (푎 < 푏) ℎ = step size (푏 − 푎 = (푛 − 1)ℎ where 푛 is a positive integer) Support 푥 ∈ {푎, 푎 + ℎ, … , 푎 + (푛 − 1)ℎ(= 푏)} Probability mass function 1 푓(푥) = 푓표푟 푥 = 푎, 푎 + ℎ, … , 푎 + (푛 − 1)ℎ 푛 Cumulative distribution 0 푓표푟 푥 < 푎 function 푥 − 푎 1 퐹(푥) = {푖푛푡 ( + ) 푓표푟 푎 ≤ 푥 < 푏 ℎ 푛 1 푓표푟 푥 ≥ 푏 Mean 푎 + 푏

2 Variance (푏 − 푎)(푏 − 푎 + 2ℎ)

12 Skewness 0 (Excess) kurtosis 6(푛2 + 1) − 5(푛2 − 1) Characteristic function 1 푒푖푎푡 − 푒푖(푏+1)푡 ( ) 푛 1 − 푒푖푡 Other comments The median of this distribution is the same as its mean.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “uniform (discrete)”. For details of other supported probability distributions see here.

Continuous (univariate) distributions

The [BetaDistribution]

The beta distribution describes a distribution in which outcomes are limited to a specific range, the probability density function within this range being characterised by two shape parameters.

Distribution name Beta distribution Common notation 푋~퐵푒푡푎(훼, 훽) Parameters 훼 = shape parameter (훼 > 0) 훽 = shape parameter (훽 > 0) Domain 0 ≤ 푥 ≤ 1 Probability density function 푥훼−1(1 − 푥)훽−1 푓(푥) = 퐵(훼, 훽) Cumulative distribution 퐵푥(훼, 훽) 퐹(푥) = = 퐼 (훼, 훽) function 퐵(훼, 훽) 푥 훼 Mean 훼 + 훽 Variance 훼훽

(훼 + 훽)2(훼 + 훽 + 1) Skewness 2(훽 − 훼)√훼 + 훽 + 1

(훼 + 훽 + 2)√훼훽 (Excess) kurtosis 6((훼 − 훽)2(훼 + 훽 + 1) − 훼훽(훼 + 훽 + 2))

훼훽(훼 + 훽 + 2)(훼 + 훽 + 3) Characteristic function ∞ 푘−1 훼 + 푟 (푖푡)푘 퐹 (훼; 훼 + 훽; 푖푡) = 1 + ∑ (∏ ) 1 1 훼 + 훽 + 푟 푘! 푘=1 푟=0 Other comments The beta distribution is also known as a beta distribution of the first 훼−1 kind. Its mode is for 훼 > 1, 훽 > 1. There is no simple closed 훼+훽−2 form solution for its median.

The beta distribution parameters are sometimes taken to include boundary parameters 푎, 푏 (푎 < 푏) in which case its domain is 푎 ≤ (푥−푎)훼−1(푏−푥)훽−1 푥 ≤ 푏, and its pdf and cdf are 푓(푥) = and 퐹(푥) = 퐵(훼,훽) 퐵 (훼,훽) 푥−푎 훼푏+훽푎 (훼−1)푏+(훽−1)푎 푧 where 푧 = , its mean is , its mode is 퐵(훼,훽) 푏−푎 훼+훽 훼+훽−2 훼훽(푏−푎)2 for 훼 > 1, 훽 > 1 and its variance is (훼+훽)2(훼+훽+1)

푋 If 푋~Γ(훼, 휃) and 푌~Γ(훽, 휃) then ~퐵푒(훼, 훽). If 푋~퐵푒푡푎(훼, 훽) 푋+푌 푛 푚 then 푋⁄(1 − 푋) ~퐵푒푡푎푃푟푖푚푒(훼, 훽) and if 푋~퐵푒푡푎 ( , ) then 2 2 푚푋 ~퐹(푛, 푚) (if 푛 > 0 and 푚 > 0). The 푘’th order statistic of a 푛(1−푋) sample of size 푛 from the uniform distribution has a beta distribution, 푢(푘)~퐵푒푡푎(푘, 푛 + 1 − 푘).

If 푋~퐵푒푡푎(1 + 휆 (푚 − 푎)⁄(푏 − 푎) , 1 + 휆 (푏 − 푚)⁄(푏 − 푎)) then 푎 + 푋(푏 − 푎)~푃푒푟푡(푎, 푏, 푚, 휆), i.e. the Pert distribution is a special case of the beta distribution.

훼+푗 Its non-central moments are 퐸(푋푟) = ∏푘−1 . 푗=0 훼+훽+푗

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “beta”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “beta4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [BetaPrimeDistribution]

Distribution name Beta prime distribution Common notation 푋~훽′(훼, 훽) Parameters 훼 = shape parameter (훼 > 0) 훽 = shape parameter (훽 > 0) Domain 푥 ≥ 0 Probability density function 푥훼−1(1 + 푥)−훼−훽 푓(푥) = 퐵(훼, 훽) Cumulative distribution 퐹(푥) = 퐼푥⁄(1+푥)(훼, 훽) function 훼 Mean 푖푓 훽 > 1 훽 − 1 Variance 훼(훼 + 훽 − 1) 푖푓 훽 > 2 (훽 − 1)2(훽 − 2) Other comments The beta prime distribution is also called the inverted beta or the beta distribution of the second kind or the Pearson Type 6 distribution. The mode of the beta prime distribution is 훼−1 for 훼 > 1, 훽 > 1. There is no simple closed form expression 훼+훽−2 for its median.

Its non-central moments (for integral 푟) are: 푘 훼 + 푗 − 1 퐵(훼 + 푟)퐵(훽 − 푟) 퐸(푋푟) = ∏ = 훽 − 푖 퐵(훼, 훽) 푗=1

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “beta prime”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “beta prime4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [BurrDistribution]

Distribution name Burr Common notation 푋~퐵푢푟푟(훼, 훽, 푘) or 푋~푆푀(훼, 훽, 푘) Parameters 푘 = shape parameter (푘 > 0) 훼 = shape parameter (훼 > 0) 훽 = shape parameter (훽 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 훼훽푘훼푥훽−1 푓(푥) = (푘 + 푥훽)훼+1 Cumulative distribution 푘 훼 퐹(푥) = 1 − ( ) function 푘 + 푥훽 Mean 1 1 푘1⁄훽 Γ (훼 − ) Γ (1 + ) 훽 훽 Γ(훼) 2 Variance 1 1 (Γ (훼 − ) Γ (1 + )) 2 2 훽 훽 푘2⁄훽 Γ (훼 − ) Γ (1 + ) − 훽 훽 Γ(훼) Γ(훼) ( ) Other comments The Burr distribution is also known as the Burr Type XII distribution or the Singh-Maddala distribution (sometimes also called the generalised log-).

Its non-central moments are: 푟 푟 푘푟⁄훽 퐸(푋푟) = Γ (훼 − ) Γ (1 + ) 푟 = 1,2, … , 푟 < 훼훽 훽 훽 Γ(훼)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “burr”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “burr5” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [CauchyDistribution]

Distribution name Cauchy distribution Common notation 푋~퐶푎푢푐ℎ푦(휇, 𝜎) Parameters 휇 = 𝜎 = (𝜎 > 0) Domain −∞ < 푥 < +∞ −1 Probability density function 푥 − 휇 2 푓(푥) = (휋𝜎 (1 + ( ) )) 𝜎 Cumulative distribution 1 푥 − 휇 1 퐹(푥) = arctan ( ) + function 휋 𝜎 2 Mean Does not exist Variance Does not exist Skewness Does not exist (Excess) kurtosis Does not exist Characteristic function exp(휇푖푡 − 𝜎|푡|) Other comments The of the Cauchy distribution is: 1 푄(푝) = 휇 + 𝜎 tan (휋 (푝 − )) 2 Its median is thus 휇.

The Cauchy distribution is a special case of the stable (more precisely the sum stable) distribution family.

The special case of the Cauchy distribution when 휇 = 0 and 𝜎 = 1 is called the standard Cauchy distribution. It coincides with the Student’s t distribution with one degree of freedom. It has a 1 probability density function of 푓(푥; 0,1) = . 휋(1+푥2)

If 푋~푁(0,1) and 푌~푁(0,1) are independent random variables then 푋 ~퐶푎푢푐ℎ푦(0,1) and this can be used to generate random variates. 푌

The Cauchy distribution is also known as the Cauchy-Lorentz or Lorentz distribution (especially amongst physicists).

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “cauchy”. For details of other supported probability distributions see here.

The Chi-squared distribution [ChiSquaredDistribution]

The chi-squared distribution with 휈 degrees of freedom is the distribution of a sum of the squares of 휈 independent standard normal random variables. A consequence is that the sum of independent chi-squared variables is also chi-squared distributed. It is widely used in hypothesis testing, goodness of fit analysis or in constructing confidence intervals. It is a special case of the .

Distribution name Chi-squared distribution 2 Common notation 푋~휒휈 Parameters 휈 = degrees of freedom (positive integer) Domain 0 ≤ 푥 < +∞ 푥 Probability density function 푥휈⁄2−1 exp (− ) 2 푓(푥) = 2휈⁄2Γ(휈⁄2) Cumulative distribution Γ푥⁄2(휈⁄2) 퐹(푥) = function Γ(휈⁄2) Mean 휈 Variance 2휈 Skewness 2 2√ 휈 (Excess) kurtosis 12

휈 Characteristic function (1 − 2푖푡)−휈⁄2 Other comments 2 3 Its median is approximately 휈 (1 − ) . Its mode is max(휈 − 2,0). Is 9휈 also known as the central chi-squared distribution (when there is a need to contrast it with the noncentral chi-squared distribution).

In the special case of 휈 = 2 the cumulative distribution function simplifies to 퐹(푥) = 1 − 푒−푥⁄2.

2 2 As 휈 → ∞, (휒휈 − 휈)⁄√2휈 → 푁(0,1) and 푞퐹(푞, 휈) → 휒푞

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “chi-squared”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “chi-squared3” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [DagumDistribution]

Distribution name Dagum distribution Common notation 푋~퐷푎푔푢푚(훼, 훽, 푘) Parameters 푘 = shape parameter (푘 > 0) 훼 = shape parameter (훼 > 0) 훽 = scale parameter (훽 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 푥 훼푘−1 훼푘 ( ) 훽 푓(푥) = 푘+1 푥 훼 훽 (1 + ( ) ) 훽 Cumulative distribution 푥훼푘 퐹(푥) = function (훽훼 + 푥훼)푘 Mean 1 1 Γ (− ) Γ ( + 푘) 훽 훼 훼 {− 훼 > 1 훼 Γ(푘) ∞ otherwise Variance 훽 2 − ( ) (퐴 + 퐵) 훼 > 2 { 훼 ∞ otherwise where 2 2 2훼Γ (− ) Γ ( + 푘) 훼 훼 퐴 = Γ(푘) 1 1 2 Γ (− ) Γ ( + 푘) 훼 훼 퐵 = ( ) Γ(푘) Other comments Also known as the Dagum type 1 distribution. Is used in modelling income distributions. The cdf of a Dagum type 2 distribution adds a point mass at the origin and then follows a Dagum type 1 distribution over the positive halfline.

−1⁄훼 훼푘−1 1⁄훼 Its median is 훽(21⁄푘 − 1) and its mode is 훽 ( ) . 훼+1

Its non-central moments are: 푟 푟 훽푟 퐸(푋푟) = Γ (− + 1) Γ ( + 푘) 푟 = 1,2, … , 푟 < 훼 훼 훼 Γ(푘)

If 푋~퐷푎푔푢푚(훼, 훽, 푘) then 1⁄푋 ~퐵푢푟푟(훼, 1⁄훽 , 푘).

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “dagum”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “dagum4” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [DegenerateDistribution]

The degenerate distribution characterises a distribution involving a single . It can be viewed as the limiting case of many common distributions in which the scale parameter tends to zero, so the distribution function concentrates onto a single point. It is also called the .

Distribution name Degenerate distribution (can also be viewed as a discrete distribution) Common notation 푋~훿푎 Parameters 푎 = location parameter Domain 푥 = 푎 Probability density function 1 푥 − 푎 2 푓(푥) = lim (exp (− ( ) )) 휎→0 2 𝜎 1, 푥 > 푎 Cumulative distribution 푓(푥) = { function 0, 푥 < 0 Mean 푎 Variance 0 Skewness Does not exist (Excess) kurtosis Does not exist Characteristic function 푒푖푡푎

Nematrian web functions

Given its degenerate form, no functions relating to the above distribution are accessible via the Nematrian web function library. For details of other supported probability distributions see here.

The error function distribution [ErrorFunctionDistribution]

The error function distribution with parameter ℎ is a special case of the normal distribution, i.e. 1 푁 (0, ). 2ℎ

The [ExponentialDistribution]

The exponential distribution describes the time between events if these events follow a Poisson process (i.e. a in which events occur continuously and independently of one another). It is also called the negative exponential distribution. It is not the same as the of distributions.

Distribution name Exponential distribution Common notation 푋~퐸푥푝(휆) Parameters 휆 = inverse scale (i.e. rate) parameter (휆 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 푓(푥) = 휆 exp(−휆푥) Cumulative distribution 퐹(푥) = 1 − exp(−휆푥) function Mean 1

휆 Variance 1

휆2 Skewness 2 (Excess) kurtosis 6 Characteristic function (1 − 푖 푡⁄휆)−1 Other comments The exponential distribution is a special case of the Gamma distribution, as if 푋~퐸푥푝(휆) then 푋~Γ(1, 1⁄휆).

The mode of an exponential distribution is 0. The quantile function, i.e. the inverse cumulative distribution function, is 퐹−1(푝; 휆) = log(1−푝) − . 휆

Γ(1+푟) The non-central moments (푟 = 1,2,3, … ) are 퐸(푋푟) = . Its 휆푟 log 2 median is . 휆

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “exponential”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “exponential2” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The F distribution [FDistribution]

The F distribution is also known as Snedecor’s F or the Fisher-Snedecor distribution. It commonly arises in statistical tests linked to analysis of variance.

Distribution name F distribution Common notation 푋~퐹(휈1, 휈2) Parameters 휈1 = degrees of freedom (first) (positive integer) 휈2 = degrees of freedom (second) (positive integer) Domain 0 ≤ 푥 < +∞ Probability density function 휈 휈 1 (휈1푥) 1휈2 2 푓(푥) = √ 휈 +휈 푥퐵(휈1⁄2 , 휈2⁄2) (휈1푥 + 휈2) 1 2

Cumulative distribution 퐵 ⁄( )(휈 ⁄2 , 휈 ⁄2) ( ) 휈1푥 휈1푥+휈2 1 2 ( ⁄ ⁄ ) 퐹 푥 = = 퐼휈1푥⁄(휈1푥+휈2) 휈1 2 , 휈2 2 function 퐵(휈1⁄2 , 휈2⁄2) Mean 휈1 for 휈2 > 2 휈2 − 2 2 Variance 2휈2 (휈1 + 휈2 − 2) 2 for 휈2 > 4 휈1(휈2 − 2) (휈2 − 4) Skewness (2휈1 + 휈2 − 2)√8(휈2 − 4) for 휈2 > 6 (휈2 − 6)√휈1(휈1 + 휈2 − 2) 2 (Excess) kurtosis 휈1(5휈2 − 22)(휈1 + 휈2 − 2) + (휈2 − 4)(휈2 − 2) 12 for 휈2 > 8 휈1(휈2 − 6)(휈2 − 8)(휈1 + 휈2 − 2) 휈 + 휈 Characteristic function Γ ( 1 2) 2 휈1 휈2 휈2 휈 푈 ( , 1 − , − 푖푡) Γ ( 2) 2 2 휈1 2 Where 푈(푎, 푏, 푧) is the confluent hypergeometric function of the second kind Other comments The F distribution is a special case of the Pearson type 6 distribution. It is also a particular example of the beta prime distribution.

2 2 If 푋1~휒 (휈1) and 푋2~휒 (휈2) are independent random variables then

푋1⁄휈1 ~퐹(휈1, 휈2) 푋2⁄휈2

(휈1−2) 휈2 Its mode is for 휈1 > 2. There is no simple closed form 휈1 휈2+2 for the median.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “f”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “f4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The Fatigue distribution [FatigueDistribution]

The fatigue distribution, also known as the Birnbaum-Saunders distribution or the fatigue life distribution is used extensively to model failure times.

Distribution name (standard) Fatigue distribution Common notation 푋~퐵푖푟푛푏푎푢푚 − 푆푎푢푛푑푒푟푠(훼) Parameters 훼 = shape parameter (훼 > 0) Domain 0 < 푥 < +∞ Probability density function √푥 + √1⁄푥 √푥 − √1⁄푥 푓(푥) = 휙 ( ) 2훼푥 훼 1 푥2 where 휙(푥) = exp (− ) √2휋 2 Cumulative distribution √푥 − √1⁄푥 퐹(푥) = 푁 ( ) function 훼 Mean 훼2 1 + 2 Variance 훼2(4 + 5훼2)

4 Other comments Its quantile function, i.e. the inverse of its cdf, is: 2 1 2 퐹−1(푝) = (훼푁−1(푝) + √4 + (훼푁−1(푝)) ) 4

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “fatigue”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “fatigue3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The Frechét distribution [FrechetDistribution]

The Frechét distribution is a special case of the generalised extreme value (GEV) distribution. It characterises the distribution of ‘block maxima’ under certain (relatively restrictive) conditions in situations where the tail index corresponds to fatter tails than arises with the normal distribution.

The gamma distribution [GammaDistribution]

The gamma distribution is a two-parameter family of continuous probability distributions. Two different parameterisations are in common use, see below, with the (푘, 휃) parameterisation being apparently somewhat more common in econometrics and the (푘, 휆) parameterisation being somewhat more common in Bayesian .

Distribution name Gamma distribution Common notation 푋~Γ(푘, 휃) or Γ(훼, 휆) Parameters Has two commonly used parameterisations: 푘 = shape parameter (푘 > 0) 휃 = scale parameter (휃 > 0) or 휆 = inverse scale (i.e. rate) parameter (휃 > 0) where 휆 = 1⁄휃. Unless otherwise specified the material below assumes the first parameterisation (i.e. using a scale parameter) Domain 0 ≤ 푥 < +∞ Probability density function 푥푘−1푒−푥⁄휃 푓(푥) = 휃푘Γ(푘) Cumulative distribution Γ푥⁄휃(푘) 퐹(푥) = function Γ(푘) Mean 푘휃 Variance 푘휃2 Skewness 2⁄√푘 (Excess) kurtosis 6⁄푘 generating (1 − 휃푡)−푘 푓표푟 푡 < 1⁄휃 function Characteristic function (1 − 푖휃푡)−푘 Other comments The gamma distribution can also be defined with a location parameter, 훾, say, in which case its domain is shifted to 훾 ≤ 푥 < +∞.

Its mode is (푘 − 1)휃 for 푘 ≥ 1.

If 푋 follows an exponential distribution with rate parameter 휆 then 푋~Γ(1, 1⁄휆).

If 푋 follows a chi-squared distribution, with 휐 degrees of freedom, i.e. i.e. 푋~휒2(휐) then 푋~Γ(휐⁄2 , 2) and 푐푋~Γ(휐⁄2 , 2푐).

If 푘 is integral then Γ(푘, 휃) is also called the . It is the distribution of the sum of 푘 independent exponential variables each with mean 휃 = 1⁄휆. Events that occur independently with some average rate are commonly modelled using a Poisson process. The waiting times between 푘 occurrences of the event are then Erlang distributed whilst the number of events in a given amount of time is Poisson distributed.

If 푋 follows a Maxwell-Boltzmann distribution with parameter 푎 then 푋2~ Γ(3⁄2 , 휃 = 2푎2) . If 푋 follows a skew logistic distribution with parameter 휃 then log(1 + 푒−푋) ~ Γ(1, 휃).

The gamma distribution is the for the precision (i.e. inverse variance) of a normal distribution and for the exponential distribution.

The gamma distribution has the ‘summation’ property that if 푋푖~Γ(푘푖, 휃) for 푖 = 1, … , 푛 and the 푋푖 are independent then 푛 푛 ∑푖=1 푋푖 ~Γ(∑푖=1 푘푖 , 휃).

Γ(푘+푟) Its non-central moments (푟 = 1,2,3, … ) are 퐸(푋푟) = 휃푟. Γ(푘) There is in general no simple closed form for its median.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “gamma”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “gamma3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The generalised extreme value distribution [GEVDistribution]

The generalised extreme value (or generalized extreme value) distribution characterises the behaviour of ‘block maxima’ under certain (somewhat restrictive) regularity conditions. See also Nematrian’s webpages about (EVT).

Distribution name Generalised extreme value (GEV) distribution (for maxima) Common notation 푋~퐺퐸푉(휉, 휇, 𝜎) Parameters 휉 = shape parameter 휇 = location parameter 𝜎 = scale parameter 푥 − 휇 Domain 1 + ( ) 휉 > 0 휉 ≠ 0 𝜎 −∞ < 푥 < ∞ 휉 = 0 Probability density function 1 푓(푥) = 푄(푥)휉+1푒−푄(푥) 𝜎 where −1⁄휉 푥 − 휇 (1 + 휉 ( )) 휉 ≠ 0 푄(푥) = 𝜎 푥 − 휇 exp (− ) 휉 = 0 { 𝜎 Cumulative distribution 퐹(푥) = 푒−푄(푥) function Mean Γ(1 − 휉) − 1 휇 + 𝜎 푖푓 휉 ≠ 0, 휉 < 1 휉

휇 + 𝜎훾 푖푓 휉 = 0 { ∞ 휉 ≥ 1 푛 1 where 훾 is Euler’s constant, i.e. lim (∑푘=1 − log 푛) 푛→∞ 푘 Variance 푔 − 푔2 𝜎2 2 1 푖푓 휉 ≠ 0, 휉 < 1⁄2 휉2 𝜎2휋2 푖푓 휉 = 0 6 { ∞ 휉 ≥ 1⁄2 Where 푔푘 = Γ(1 − 푘휉) Skewness 푔 − 3푔 푔 + 2푔3 3 1 2 1 푖푓 휉 ≠ 0 (푔 − 푔2)3⁄2 2 1 12√6𝜍(3) 푖푓 휉 = 0 { 휋3 1 where 𝜍(푥) is the Riemann zeta function, i.e. ∑∞ . 푘=1 푘푥 (Excess) kurtosis 푔 − 4푔 푔 + 6푔 푔2 − 3푔4 4 1 3 2 1 1 푖푓 휉 ≠ 0 ( 2)2 푔2 − 푔1 12 푖푓 휉 = 0 { 5 Other comments 휉 defines the tail behaviour of the distribution. The sub-families defined by 휉 = 0 (Type I), 휉 > 0 (Type II) and 휉 < 0 (Type III) correspond to the Gumbel, Frechét and Weibull families respectively.

An important special case when analysing threshold exceedances involves 휇 = 0 (and normally 휉 > 0) and this special case may be referred to as 퐺퐸푉(휉, 𝜎).

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “gev”. For details of other supported probability distributions see here.

The generalised gamma distribution [GeneralisedGammaDistribution]

The generalised gamma (or generalized gamma) distribution is a generalisation of the gamma distribution that includes several of the parametric distributions typically used in survival analysis.

Distribution name Generalised gamma distribution Common notation 푋~퐺푒푛푒푟푎푙푖푠푒푑퐺푎푚푚푎(푘, 훼, 훽) Parameters 푘 = shape parameter (푘 > 0) 훼 = shape parameter (훼 > 0) 훽 = scale parameter (훽 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 푘 푥 푘훼−1 푥 푘 푓(푥) = ( ) exp (− ( ) ) 훽Γ(훼) 훽 훽

Cumulative distribution Γ(푥⁄훽)푘 (훼) 퐹(푥) = function Γ(훼) 1 Mean Γ (훼 + ) 훽 푘 Γ(훼) Variance 2 1 2 Γ (훼 + ) Γ(훼) − Γ (훼 + ) 훽2 푘 푘 Γ(훼)2 Skewness 3 2 1 1 3 Γ (훼 + ) Γ(훼)2 − 3Γ (훼 + ) Γ (훼 + ) Γ(훼) + 2Γ (훼 + ) 푘 푘 푘 푘 3⁄2 2 1 2 (Γ (훼 + ) Γ(훼) − Γ (훼 + ) ) 푘 푘 (Excess) kurtosis 퐴 + 퐵 + 퐶 + 퐷 2 − 3 2 1 2 (Γ (훼 + ) Γ(훼) − Γ (훼 + ) ) 푘 푘 where: 4 퐴 = Γ (훼 + ) Γ(훼)3 푘 3 1 퐵 = −4Γ (훼 + ) Γ (훼 + ) Γ(훼)2 푘 푘 2 1 2 퐶 = 6Γ (훼 + ) Γ (훼 + ) Γ(훼) 푘 푘 1 4 퐷 = −3Γ (훼 + ) 푘 Other comments Its non-central moments are: 푟 훽푟Γ (훼 + ) 퐸(푋푟) = 푘 Γ(훼) The is a special case with 훼 = 1. The gamma distribution is a special case with 푘 = 1.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “generalised gamma”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “generalised gamma4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The generalised inverse Gaussian distribution [GeneralisedInverseGaussianDistribution]

Distribution name Generalised inverse Gaussian distribution (GIG) Common notation 푋~퐺퐼퐺(휆, 휉, 휓) Parameters 휆 = parameter (휆 > 0) 휉 = parameter (휉 ≥ 0) 휓 = parameter (휓 ≥ 0) Domain 0 < 푥 < +∞ Probability density function 휆 (√휓⁄휉) 1 휉 푓(푥) = 푥휆−1 exp (− ( + 휓푥)) 2퐾휆(√휉휓) 2 푥 where 퐾휆(푥) is the modified Bessel function of the third kind with index 휆.

Mean √휉퐾푝+1(√휉휓)

√휓퐾푝(√휉휓) Variance 2 휉 퐾푝+2(√휉휓) 퐾푝+1(√휉휓) ( ) ( − ( ) ) 휓 퐾푝(√휉휓) 퐾푝(√휉휓)

푝⁄2 Characteristic function 휓 퐾푝(√휉(휓 − 2푖푡)) ( ) 휓 − 2푖푡 퐾푝(√휉휓) Other comments When 휉 > 0 and 휓 > 0 the non-central moments are: 푟⁄2 휉 퐾휆+푟(√휉휓) 퐸(푋푟) = ( ) 휓 퐾휆(√휉휓)

Some commentators use GIG to refer to the generalised integer gamma distribution (which is not the same as the generalised inverse Gaussian distribution).

Nematrian web functions

This distribution is not currently supported within the Nematrian web function library. For details of other supported probability distributions see here.

The generalised [GeneralisedParetoDistribution]

The generalised Pareto distribution (generalized Pareto distribution) arises in Extreme Value Theory (EVT). If the relevant regularity conditions are satisfied then the tail of a distribution (above some suitably high threshold), i.e. the distribution of ‘threshold exceedances’, tends to a generalized Pareto distribution.

Care is needed with EVT because what we are in effect doing with it is to extrapolate into the tail of the distribution. Extrapolation is an intrinsically imprecise and subjective mathematical activity. We can in effect view the regularity conditions that need to be satisfied if EVT applies as corresponding to requiring that this extrapolation is done in a particular manner.

Distribution name Generalised Pareto distribution (GPD) Common notation 푋~퐺푃퐷(휉, 휇, 𝜎) Parameters 휉 = shape parameter 휇 = location parameter 𝜎 = scale parameter (𝜎 > 0) Domain 휇 ≤ 푥 < +∞ 휉 ≥ 0 𝜎 휇 ≤ 푥 ≤ 휇 − 휉 < 0 휉 Probability density function 1 (1 + 휉푧)−1−1⁄휉 휉 ≠ 0 푓(푥) = {𝜎 1 exp(−푧) 휉 = 0 𝜎 where 푥 − 휇 푧 = 𝜎 Cumulative distribution 1 − (1 + 휉푧)−1⁄휉 휉 ≠ 0 퐹(푥) = { function 1 − exp(−푧) 휉 = 0 𝜎 Mean 휇 + 휉 < 1 1 − 휉 Variance 𝜎2 1 휉 < (1 − 2휉)(1 − 휉)2 2 Skewness 2(1 + 휉)√1 − 2휉 1 휉 < 1 − 3휉 3 (Excess) kurtosis 6(1 + 휉 − 6휉2 − 2휉3) 1 휉 < (1 − 3휉)(1 − 4휉) 4 Other comments If 푋 is uniformly distributed, 푋~푈(0,1) then the variable 푌 = 휇 + 𝜎(푋−휉 − 1)⁄휉 ~퐺푃퐷(휇, 𝜎, 휉).

The mean excess function for a GPD, i.e. 푒(푢) = 퐸(푋 − 푢|푋 > 푢) takes a particularly simple form which is linear in 휉, i.e. 𝜎 휉(푢 − 휇) 푒(푢) = + 1 − 휉 1 − 휉

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “generalised pareto”. For details of other supported probability distributions see here.

The [GumbelDistribution]

The Gumbel distribution is a special case of the generalised extreme value distribution. It characterises the distribution of ‘block maxima’ as per Extreme Value Theory (EVT) under certain (relatively restrictive) conditions.

The hyperbolic secant distribution

[Nematrian website page: HyperbolicSecantDistribution, © Nematrian 2015]

Distribution name Hyperbolic secant distribution Common notation 푋~푆푒푐ℎ(휇, 𝜎) Parameters 𝜎 = scale parameter (𝜎 > 0) 휇 = location parameter Domain −∞ < 푥 < +∞ 휋 Probability density function sech ( 푧) 2 푓(푥) = 2𝜎 where 푥 − 휇 푧 = 𝜎 Cumulative distribution 2 휋 퐹(푥) = arctan (exp ( 푧)) function 휋 2 Mean 휇 Variance 𝜎2 Skewness 0 (Excess) kurtosis 2 Characteristic function 푒−푖푡휇 sech(𝜎푡)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “hyperbolic secant”. For details of other supported probability distributions see here.

The inverse gamma distribution [InverseGammaDistribution]

The inverse gamma distribution describes the distribution of the reciprocal of a variable distributed according to the gamma distribution.

Distribution name Inverse gamma Common notation Parameters 훼 = shape parameter (훼 > 0) 훽 = scale parameter (훽 > 0) Domain 0 < 푥 < +∞ 훽 Probability density function exp (− ) 푥 푓(푥) = 훽Γ(훼)(푥⁄훽)훼+1 Cumulative distribution Γ훽⁄푥(훼) 퐹(푥) = 1 − function Γ(훼) Mean 훽 푓표푟 훼 > 1 훼 − 1 Variance 훽2 푓표푟 훼 > 2 (훼 − 1)2(훼 − 2) Skewness 4√훼 − 2 푓표푟 훼 > 3 훼 − 3 (Excess) kurtosis 30훼 − 66 푓표푟 훼 > 4 (훼 − 3)(훼 − 4) Characteristic function 2(−푖훽푡)훼⁄2 퐾 (2√−푖훽푡) Γ(훼) 훼 Other comments Also called the log Pearson type 5 distribution.

Its mode is 훽⁄(훼 + 1)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “inverse gamma”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “inverse gamma3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The inverse Gaussian distribution [InverseGaussianDistribution]

While the Gaussian (i.e. normal distribution describes a Brownian motion’s level at a fixed time, the inverse Gaussian distribution describes the distribution of time a Brownian motion starting at 0 with positive drift takes to reach a fixed positive level.

Distribution name Inverse Gaussian distribution Common notation 푋~퐼퐺(휆, 휇) Parameters 휆 = parameter (휆 > 0) 휇 = parameter (휇 > 0) Domain 0 < 푥 < +∞ Probability density function 휆 휆(푥 − 휇)2 푓(푥) = √ exp (− ) 2휋푥3 2휇2푥 Cumulative distribution 휆 푥 휆 푥 2휆 function 퐹(푥) = 푁 (√ ( − 1)) + 푁 (−√ ( + 1)) exp ( ) 푥 휇 푥 휇 휇

Mean 휇 Variance 휇3⁄휆 Skewness 휇 1⁄2 3 ( ) 휆 (Excess) kurtosis 15휇

휆 Characteristic function 휆 2휇2푖푡 exp ( (1 − √1 − )) 휇 휆

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “inverse gaussian”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “inverse gaussian3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The Johnson SU distribution [JohnsonSUDistribution]

Distribution name Johnson SU distribution Common notation 푋~퐽표ℎ푛푠표푛푆푈(훾, 훿, 휆, 휉) or 푋~푆푈(훾, 훿, 휆, 휉) Parameters 훾 = shape parameter 훿 = shape parameter (훿 > 0) 휆 = location parameter 휉 = scale parameter (휉 > 0) Domain −∞ < 푥 < +∞ 1 Probability density function 훿 exp (− (훾 + 훿 sinh−1 푧)2) 2 푓(푥) = 휉√2휋√푧2 + 1 where 푥 − 휆 푧 = 휉 Note sinh−1 푧 = log(푧 + √푧2 + 1) Cumulative distribution 퐹(푥) = 푁(훾 + 훿 sinh−1 푧) function Mean 휆 − 휉√푤 sinh Ω where 푤 = exp(훿−2) and Ω = 훾⁄훿 Variance 휉2 (푤 − 1)(푤 cosh(2Ω) + 1) 2 Skewness 휉3√푤(푤 − 1)2(푤(푤 + 2) sinh(3Ω) + 3 sinh Ω) − 4𝜎3 휉2 where 𝜎 = √ (푤 − 1)(푤 cosh(2Ω) + 1) 2 (Excess) kurtosis 휉4(푤 − 1)2(퐴 + 퐵 + 퐶) − 3 8𝜎4 where 퐴 = 푤2(푤4 + 2푤3 + 3푤2 − 3) cosh(4Ω) 퐵 = 4푤2(푤 + 2) cosh(2Ω) 퐶 = 3(2푤 + 1)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “johnsonsu”. For details of other supported probability distributions see here.

The [KumaraswamyDistribution]

The Kumaraswamy distribution describes a distribution in which outcomes are limited to a specific range, the probability density function within this range being characterised by two shape parameters. It is similar to the beta distribution but possibly easier to use because it has simpler analytical expressions for its probability density function and cumulative distribution function.

Distribution name Kumaraswamy distribution Common notation 푋~퐾푢푚푎푟푎푠푤푎푚푦(훼, 훽) Parameters 훼 = shape parameter (훼 > 0) 훽 = shape parameter (훽 > 0) Domain 0 ≤ 푥 ≤ 1 Probability density function 푓(푥) = 훼훽푥훼−1(1 − 푥훼)훽−1 Cumulative distribution 퐹(푥) = 1 − (1 − 푥훼)훽 function Mean 1 훽퐵 (1 + , 훽) 훼 Variance 2 1 2 (훽퐵 (1 + , 훽) − 훽2퐵 (1 + , 훽) ) 훼 훼 Other comments A standard Kumaraswamy distribution has 푎 = 0 and 푏 = 1. Its non- central moments are given by: 훽Γ(1 + 푟⁄훼)Γ(훽) 푟 퐸(푋푟) = = 훽퐵 (1 + , 훽) Γ(1 + 훽 + 푟⁄훼) 훼 1⁄훼 훼−1 1⁄훼 and its median is (1 − 2−1⁄훽) and its mode is ( ) (for 훼 ≥ 훼훽−1 1,훽 ≥ 1, (훼, 훽) ≠ (1,1).

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “kumaraswamy”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “kumaraswamy4” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [LaplaceDistribution]

The Laplace distribution is akin to two exponential distributions spliced together.

Distribution name Laplace distribution Common notation 푌~퐿푎푝푙푎푐푒(휇, 푏) Parameters 휇 = location parameter 푏 = scale parameter (푏 > 0) Domain −∞ < 푥 < +∞ Probability density function 1 |푥 − 휇| 푓(푥) = exp (− ) 2푏 푏 Cumulative distribution 1 푥 − 휇 exp ( ) 푥 ≤ 휇 function 퐹(푥) = { 2 푏 1 푥 − 휇 1 − exp (− ) 푥 > 휇 2 푏 Mean 휇 Variance 2푏2 Skewness 0 (Excess) kurtosis 3 Characteristic function 푒푖휇푡

1 + 푏2푡2 Other comments The median and mode are 휇. The inverse cdf (i.e. quantile) function is 1 1 퐹−1(푝) = 휇 − 푏 sgn (푝 − ) log (1 − 2 |푝 − |). 2 2

1 1 If 푋~푈 (− , ) and 푌 = 휇 − 푏 sgn(푋) log(1 − 2|푋|) then 2 2 푌~퐿푎푝푙푎푐푒(휇, 푏).

It is sometimes referred to as the double exponential distribution (but this term is also apparently also used sometimes of the Gumbel distribution)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “laplace”. For details of other supported probability distributions see here.

The Lévy distribution [LevyDistribution]

The time of hitting a single point 푎 (different from the starting point of 0) of a Brownian motion without a mean drift follows the Lévy distribution with 𝜎 = 푎2. With a mean drift it follows an inverse Gaussian meaning that the Lévy distribution is a special case of the inverse Gaussian distribution.

Distribution name Lévy distribution Common notation 푋~퐿푒푣푦(𝜎) Parameters 𝜎 = scale parameter (𝜎 > 0) Domain 0 < 푥 < +∞ 𝜎 Probability density function exp ( ) 1 2푥 푓(푥) = 𝜎√2휋 (푥⁄𝜎)3⁄2 Cumulative distribution 𝜎 퐹(푥) = 2 − 2푁 (√ ) function 푥 Mean ∞ Variance ∞ Skewness undefined (Excess) kurtosis undefined Characteristic function 푒−√−2푖휎푡

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “levy”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “levy2”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The logistic distribution [LogisticDistribution]

The logistic distribution has a similar shape to the normal distribution but has heavier tails.

Distribution name Logistic distribution Common notation 푋~퐿표푔푖푠푡푖푐(휇, 푠) Parameters 휇 = location parameter 푠 = scale parameter (𝜎 > 0) Domain −∞ < 푥 < +∞ Probability density function exp(−푧) 푓(푥) = 푠(1 + exp(−푧))2 where 푥 − 휇 푧 = 푠 Cumulative distribution 1 퐹(푥) = function 1 + exp(−푧) Mean 휇 Variance 휋2푠2

3 Skewness 0 (Excess) kurtosis 6

5 Characteristic function 푒휇푖푡휋푠푡

sinh(휋푠푡) Other comments The logistic distribution is sometimes called the sech-square(d) distribution because its pdf can also be expressed in terms of the hyperbolic secan function: exp(−푧) 1 푧 푓(푥) = = sech2 ( ) 𝜎(1 + exp(−푧))2 4𝜎 2 It should not then be confused with the hyperbolic secant distribution.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “logistic”. For details of other supported probability distributions see here.

The log-logistic distribution [LogLogisticDistribution]

Distribution name Log-logistic distribution Common notation 푋~퐿퐿(훼, 훽) Parameters 훼 = scale parameter (훼 > 0) 훽 = shape parameter (훽 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 훽 푥 훽−1 ( ) ( ) 훼 훼 푓(푥) = 2 푥 훽 (1 + ( ) ) 훼 Cumulative distribution 1 퐹(푥) = function 푥 −훽 1 + ( ) 훼 Mean 훼푏 csc(푏) 푖푓 훽 > 1 where 푏 = 휋⁄훽 and csc 푥 = 1⁄sin 푥 is the cosecant function Variance 훼2(2푏 csc(2푏) − 푏2 csc2 푏) 푖푓 훽 > 2 Other comments Is also called the Fisk distribution. Its non-central moments are (if 푘 < 훽): 훼푘푘푏 퐸(푋푟) = 훼푘퐵(1 − 푘⁄훽 , 1 + 푘⁄훽) = sin(푘푏)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “loglogistic”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “loglogistic3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The lognormal distribution [LognormalDistribution]

Distribution name Lognormal distribution Common notation 푋~푙표푔푁(휇, 𝜎2) Parameters 𝜎 = scale parameter (𝜎 > 0) 휇 = location parameter Domain 0 < 푥 < +∞ Probability density function 1 log 푥 − 휇 2 exp − ( ) ( 2 𝜎 ) 푓(푥) = 푥𝜎√2휋 Cumulative distribution log 푥 − 휇 1 1 log 푥 − 휇 퐹(푥) = 푁 ( ) = + 푒푟푓 ( ) function 𝜎 2 2 √2𝜎2 2 Mean 푒휇+휎 ⁄2 2 2 Variance (푒(휎 ) − 1)푒2휇+휎 2 Skewness (푒(휎 ) + 2)√푒(휎2) − 1 2 2 2 (Excess) kurtosis 푒(4휎 ) + 2푒(3휎 ) + 3푒(2휎 ) − 6 Characteristic function No simple expression that is not divergent 2 Other comments The median of a lognormal distribution is 푒휇 and its mode is 푒휇−휎 .

The truncated moments of 푙표푔푁(휇, 𝜎2) are: 푈 2 2 log 푈 − 휇 ∫ 푥푘푓(푥)푑푥 = 푒푘휇+푘 휎 ⁄2 (푁 ( − 푘𝜎) 𝜎 퐿 log 퐿 − 휇 − 푁 ( − 푘𝜎)) 𝜎

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “lognormal”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “lognormal4” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [LognormalDistribution]

Distribution name Nakagami distribution Common notation 푋~푁푎푘푎푔푎푚푖(푚, ω) Parameters 푚 = parameter (푚 ≥ 1⁄2) ω = parameter (ω > 0) Domain 0 ≤ 푥 < +∞ Probability density function 2푚푚 푚 푓(푥) = 푥2푚−1 exp (− 푥2) Γ(푚)ω푚 ω Cumulative distribution Γ푚푥2⁄ω(푚) 퐹(푥) = function Γ(푚) Mean 1 Γ (푚 + ) 1⁄2 2 휔 ( ) Γ(푚) 푚 Variance 1 2 Γ (푚 + ) 1 2 휔 (1 − ( ) ) 푚 Γ(푚)

Other comments 1 (2푚−1)휔 1⁄2 Its median is √휔 and its mode is ( ) . √2 푚

If 푋~Γ(푘, 휃) then 푌 = √푋~푁푎푘푎푔푎푚푖(푘, 푘휃)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “nakagami”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “nakagami3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The non-central chi-squared distribution [NoncentralChiSquaredDistribution]

Distribution name Non-central chi-squared distribution Common notation 푋~휒2(휐, 휆) Parameters 휈 = degrees of freedom (positive integer) 휆 = non-centrality parameter (휆 ≥ 0) Domain 0 ≤ 푥 < +∞ 푥 Probability density function 푥휈⁄2−1 exp (− ) 2 푓(푥) = 퐹 (; 푘⁄2 ; 휆푥⁄4) 2휈⁄2Γ(휈⁄2) 0 1

Cumulative distribution 퐹(푥) = 1 − 푄휈⁄2(√휆, √푥) function where 푄푀(푎, 푏) is the Marcum-Q function. Mean 휈 + 휆 Variance 2(휈 + 2휆) Skewness 2√2(휈 + 3휆) 8 √ (휈 + 2휆)3⁄2 휈 (Excess) kurtosis 12(휈 + 4휆)

(휈 + 2휆)2 Characteristic function 푖휆푡 (1 − 2푖푡)−휈⁄2 exp (− ) 1 − 2푖푡 Other comments The non-central chi-squared distribution with 휈 degrees of freedom and non-centrality parameter 휆 is the distribution of the sum of the squares of 휈 independent normal distributions each with unit 휈 2 standard deviation but with non-zero means 휇푖 where 휆 = ∑푖=1 휇푖 . The (standard) chi-squared distribution is a special case of it with 휆 = 0.

Nematrian web functions

This distribution is not currently supported within the Nematrian web function library. For details of other supported probability distributions see here.

The non-central t distribution [LognormalDistribution]

Distribution name (Standard) non-central t distribution (NCT) Common notation 푋~푁퐶푇(휈, 푑) Parameters 휐 = degrees of freedom (휈 > 0, usually 휈 is an integer although in some situations a non-integral 휈 can arise) 푏 = non-centrality parameter Domain −∞ < 푥 < +∞ 휐 Probability density function (퐹 (푥√1 + 2⁄휐) − 퐹 (푥)) 푖푓 푥 ≠ 0 푥 휐+2,푏 휐,푏 푓(푥) = Γ((휐 + 1)⁄2) 푏2 exp (− ) 푖푓 푥 = 0 { √휋휐Γ(휐⁄2) 2 where 퐹휐,푑(푥) is the NCT cumulative distribution function Cumulative distribution 푄휐,푏(푥) 푖푓 푥 ≥ 0 퐹휐,푏(푥) = { function 1 − 푄휐,−푏(−푥) 푖푓 푥 < 0 where ∞ 1 1 휐 휐 푄 (푥) = 푁(−푏) + ∑ (푝 퐼 (푗 + , ) + 푞 퐼 (푗 + 1, )) 휐,푑 2 푗 푦 2 2 푗 푦 2 푗=0 푥2 푦 = 푥2 + 휐 푗 1 푏2 푏2 푝 = ( ) exp (− ) 푗 푗! 2 2 푗 휇 푏2 푏2 푞 = ( ) exp (− ) 푗 3 2Γ (푗 + ) 2 2 √ 2 휈 − 1 Mean Γ ( ) 휈 2 푏√ 휈 푖푓 휈 > 1 2 Γ ( ) 2 Variance 휈 − 1 2 2 2 Γ ( ) 휈(1 + 푏 ) 푏 휈 2 − ( 휈 ) 푖푓 휈 > 2 휈 − 2 2 Γ ( ) 2 Other comments 푋 has a standard non-central 푡 distribution with 휐 degrees of freedom and non-centrality parameter 푑 if 푋 = (푍 + 푏)⁄√푌⁄휈, 2 푌~휒휈 , 푍~푁(0,1) and 푌 and 푍 are independent. It has the following non-central moments: 휈 − 푘 푘 Γ ( ) 푘 푏2⁄2 푘 휈 2 2 −푏2⁄2 푑 푒 퐸(푋 ) = ( ) 휈 푒 푖푓 휈 > 푘 2 Γ ( ) 푑푏 2

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “non-central t”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “non-central t4”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The normal distribution [NormalDistribution]

The normal distribution is a continuous probability distribution that has a bell-shaped probability density function:

1 1 푥 − 휇 2 푓(푥) = exp (− ( ) ) 𝜎√2휋 2 𝜎

It is usually considered to be the most prominent probability distribution in statistics partly because it arises in a very large number of contexts as a result of the and partly because it is relatively tractable analytically.

The normal distribution is also called the Gaussian distribution. The unit normal (or standard normal) distribution is 푁(0,1).

Characteristics of the normal distribution are set out below:

Distribution name Normal distribution Common notation 푋~푁(휇, 𝜎2) Parameters 𝜎 = scale parameter (𝜎 > 0) 휇 = location parameter Domain −∞ < 푥 < +∞ Probability density function 1 1 푥 − 휇 2 푓(푥) ≡ 휙(푥) = exp (− ( ) ) 𝜎√2휋 2 𝜎 푥 Cumulative distribution 1 1 푡 − 휇 2 function 퐹(푥) ≡ 푁(푥) = Φ(푥) = ∫ exp (− ( ) ) 푑푡 √2휋 2 𝜎 −∞ Mean 휇 Variance 𝜎2 Skewness 0 (Excess) kurtosis 0 1 Characteristic function 푖푡휇− 휎2푡2 푒 2 Other comments The inverse unit normal distribution function (i.e. its quantile function) is commonly written 푁−1(푥) (also in some texts Φ(푥) and the unit normal density function is commonly written 휙(푥). 푁−1(푥) is also called the probit function.

1 The error function distribution is 푁 (0, ), where ℎ is now an inverse 2ℎ scale parameter ℎ > 0.

The median and mode of a normal distribution are 휇.

The truncated first moments of 푁(휇, 𝜎2) are:

푈 푈 − 휇 퐿 − 휇 ∫ 푥푓(푥)푑푥 = 휇 (푁 ( ) − 푁 ( )) 𝜎 𝜎 퐿 푈 − 휇 퐿 − 휇 − 𝜎 (휙 ( ) − 휙 ( )) 𝜎 𝜎

where 휙(푥) and 푁(푥) are the pdf and cdf of the unit normal distribution respectively.

The mean excess function of a standard normal distribution is thus 1 1 푒푥푝 (− 푢2) − 푢푁(−푢) 휙(푢) − 푢푁(−푢) √2휋 2 푒(푢) = = 푁(−푢) 푁(−푢)

The central moments of the normal distribution are: 0 푖푓 푘 푖푠 표푑푑 퐸((푋 − 휇)푘) = { 𝜎푝 × 1 × 3 × … × (푘 − 1) 푖푓 푘 푖푠 푒푣푒푛

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “normal”. For details of other supported probability distributions see here.

The Pareto distribution [ParetoDistribution]

Distribution name Pareto distribution Common notation 푋~푃푎푟푒푡표(훼, 훽) Parameters 훼 = shape parameter (훼 > 0) 훽 = scale parameter (훽 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 훼훽훼 푓(푥) = (훽 + 푥)훼+1 Cumulative distribution 훽 훼 퐹(푥) = 1 − ( ) function 훽 + 푥 Mean 훽 푓표푟 훼 > 1 훼 − 1 Variance 훼훽2 푓표푟 훼 > 2 (훼 − 1)2(훼 − 2) Skewness 2(훼 + 1) 훼 − 2 √ 푓표푟 훼 > 3 훼 − 3 훼 (Excess) kurtosis 6(훼3 + 훼2 − 6훼 − 2) 푓표푟 훼 > 4 훼(훼 − 3)(훼 − 4) Other comments Also known as the Pareto distribution of the second kind in which case the Pareto distribution of the first kind has 훽 ≤ 푥 < +∞, 훼훽훼 훽 훼 푓(푥) = and 퐹(푥) = 1 − ( ) . 푥훼+1 푥

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “pareto”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “pareto3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The power function distribution [PowerFunctionDistribution]

Distribution name Power function distribution Common notation Parameters 푘 = shape parameter (푘 > 0) Domain 0 ≤ 푥 ≤ 1 Probability density function 푓(푥) = 푘푥푘−1 Cumulative distribution 퐹(푥) = 푥푘 function Mean 푘

푘 + 1 Variance 푘

푘3 + 4푘2 + 5푘 + 2 Other comments Its non-central moments around 푎 (which can be used to derive its non-central moments around 0) are: 푘 퐸(푋푟) = 푟 + 푘

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “power”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “power3” ”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [RayleighDistribution]

Distribution name Rayleigh distribution Common notation 푋~푅푎푦푙푒푖푔ℎ(푘) Parameters 푘 = scale parameter (푘 > 0) Domain 0 ≤ 푥 < +∞ Probability density function 푥2 푥 exp (− ) 2푘2 푓(푥) = 푘2 Cumulative distribution 푥2 퐹(푥) = 1 − exp (− ) function 2푘2 Mean 휋 푘√ 2 Variance 4 − 휋 푘2 2 Skewness 2√휋(휋 − 3)

(4 − 휋)2 (Excess) kurtosis 6휋2 − 24휋 + 16 − (4 − 휋)2 Characteristic function 푘2푡2 휋 푖푘푡 1 − 푘푡 exp (− ) √ (−푖 erf ( ) − 푖) 2 2 √2 Here erf(푧) is the error function, which is defined as: 푧 2 2 erf(푧) = ∫ 푒−푡 푑푡 ⟹ erf(푥) = 2. 푁(푥√2) − 1 √휋 0 Other comments The Rayleigh distribution often arises when the overall magnitude of a vector is related to its directional components

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “rayleigh”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “rayleigh2”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [ReciprocalDistribution]

Distribution name Reciprocal distribution Common notation Parameters 푎, 푏 = boundary parameters (0 < 푎 < 푏) Domain 푎 ≤ 푥 ≤ 푏 Probability density function 1 푓(푥) = 푥(log 푏 − log 푎) Cumulative distribution log 푥 − log 푎 퐹(푥) = function log 푏 − log 푎 Mean 푏 − 푎

log 푏 − log 푎 Variance 1 푏2 − 푎2 푏 − 푎 2 ( ) − ( ) 2 log 푏 − log 푎 log 푏 − log 푎 Other comments Its non-central moments are: 푏푟 − 푎푟 퐸(푋푟) = 푟(log 푏 − log 푎)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “reciprocal”. For details of other supported probability distributions see here.

The [RiceDistribution]

The Rice distribution is characterises the magnitude of a circular bivariate normal random vector with potentially non-zero mean.

Distribution name Rice distribution Common notation 푋~푅푖푐푒(휈, 𝜎) Parameters 휈 = parameter 𝜎 = parameter Domain 0 ≤ 푥 < +∞ Probability density function 푥 푥2 + 휈2 푥휈 푓(푥) = exp (− ) 퐼 ( ) 𝜎2 2𝜎2 0 𝜎2 where 퐼0(푞) is the modified Bessel function of the first kind of order 푞 2푘 ( ) zero, i.e. 퐼 (푞) = ∑∞ 2 0 푘=0 (푘!)2 Cumulative distribution 휈 푥 퐹(푥) = 1 − 푄1 ( , ) function 𝜎 𝜎 where 푄1(푝, 푞) is the Marcum Q-function, i.e. 푄1(푝, 푞) = ∞ 푥2+푝2 ∫ 푥 exp (− ) 퐼 (푝푥)푑푥. 푞 2 0 Mean 휈2 𝜎√휋⁄2 퐿 (− ) 1⁄2 2𝜎2 2 Variance 휋𝜎2 휈2 2𝜎2 + 휈2 − (퐿 (− )) 2 1⁄2 2𝜎2 Other comments The Rayleigh distribution is a special case of the Rice distribution where 휈 = 0.

Its non-central moments are: 푟 휈2 퐸(푋푟) = 𝜎푟2푟⁄2Γ (1 + ) 퐿 (− ) 푟 = 1,2, … 2 푟⁄2 2𝜎2

where 퐿푞(푥) = 1퐹1(−푞; 1; 푥) is a Laguerre polynomial.

Nematrian web functions

This distribution is not currently supported within the Nematrian web function library. For details of supported probability distributions see here.

The Student’s t distribution [StudentsTDistribution]

The Student’s t distribution (more simply the t distribution) arises when estimating the mean of a normally distributed population when sample sizes are small and the population standard deviation is unknown.

Distribution name (Standard) Student’s t distribution Common notation 푋~푡휈 Parameters 휈 = degrees of freedom (휈 > 0, usually 휈 is an integer although in some situations a non-integral 휈 can arise) Domain −∞ < 푥 < +∞ 휈+1 휈+1 Probability density function 휈 + 1 − − Γ ( ) 2 2 2 2 1 2 푥 1 푥 푓(푥) = (1 + ) = (1 + ) 휈 1 휈 √휋휈 Γ ( ) 푣 휈퐵 ( , ) 푣 2 √ 2 2 Cumulative distribution 1 휈 1 퐼 ( , ) 푥 < 0 function 푧 퐹(푥) = { 2 2 2 1 휈 1 1 − 퐼 ( , ) 푥 ≥ 0 2 푧 2 2 where 푧 = 휈⁄(휈 + 푥2) Mean 0 휈 Variance 푓표푟 휈 > 2 휈 − 2 Skewness 0 푓표푟 휈 > 3 (Excess) kurtosis 3(휈 − 2) 6 − 3 = 푓표푟 휈 > 4 휈 − 4 휈 − 4 Characteristic function 휈⁄2 퐾휈⁄2(√휈|푡|)(√휈|푡|)

Γ(휈⁄2)2휈⁄2−1 where 퐾휈⁄2(푥) is a Bessel function Other comments It is a special case of the generalised .

Its non-central moments if 푟 is even and 0 < 푟 < 휈 are: 푟 + 1 휈 − 푟 Γ ( ) Γ ( ) 휈푟⁄2 푟 2 2 퐸(푋 ) = 휈 휋Γ ( ) √ 2

If 푟 is even and 0 < 휈 ≤ 푟 then 퐸(푋푟) = ∞, if 푟 is odd and 0 < 푟 < 휈 then 퐸(푋푟) = 0 and if 푟 is odd and 0 < 휈 ≤ 푟 then 퐸(푋푟) is undefined.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “student's t”. Functions relating to a generalised version of this distribution including additional location (i.e. shift) and scale parameters may be accessed by using a DistributionName of “student's t3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

The [TriangularDistribution]

Distribution name Triangular distribution Common notation 푋~푇푟푖푎푛푔푢푙푎푟(푎, 푏, 푐) Parameters 푎, 푏 = boundary parameters (푎 < 푏) 푐 = mode parameter (푎 ≤ 푐 ≤ 푏) Domain 푎 ≤ 푥 ≤ 푏 Probability density function 2(푥 − 푎) 푎 ≤ 푥 ≤ 푐 (푐 − 푎)(푏 − 푎) 푓(푥) = 2(푏 − 푥) 푐 < 푥 ≤ 푏 {(푏 − 푐)(푏 − 푎) Cumulative distribution (푥 − 푎)2 푎 ≤ 푥 ≤ 푐 function (푐 − 푎)(푏 − 푎) 퐹(푥) = (푏 − 푥)2 1 − 푐 < 푥 ≤ 푏 { (푏 − 푐)(푏 − 푎) Mean 푎 + 푏 + 푐

3 Variance 푎2 + 푏2 + 푐2 − 푎푏 − 푎푐 − 푏푐

18 Skewness √2(푎 + 푏 − 2푐)(2푎 − 푏 − 푐)(푎 − 2푏 + 푐)

5(푎2 + 푏2 + 푐2 − 푎푏 − 푎푐 − 푏푐)3⁄2 (Excess) kurtosis 3 − 5 Characteristic function (푏 − 푐)푒푖푎푡 − (푏 − 푎)푒푖푐푡 + (푐 − 푎)푒푖푏푡 −2 (푏 − 푎)(푐 − 푎)(푏 − 푐)푡2 Other comments Its mode is 푐. Its median is: (푐 − 푎)(푏 − 푎) 푎 + 푏 푎 + √ 푐 ≥ 2 2

(푐 − 푎)(푏 − 푎) 푎 + 푏 푏 − √ 푐 ≤ { 2 2

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “triangular”. For details of other supported probability distributions see here.

The uniform distribution [UniformDistribution]

The uniform distribution describes a (continuous) probability distribution in which any outcome within a given range is equally probable.

Distribution name Uniform distribution Common notation 푋~푈(푎, 푏) Parameters 푎, 푏 = boundary parameters (푎 < 푏) Domain 푎 ≤ 푥 ≤ 푏 Probability density function 1 푓(푥) = 푏 − 푎 푥 − 푎 Cumulative distribution 퐹(푥) = function 푏 − 푎 Mean (푎 + 푏)⁄2 Variance (푏 − 푎)2⁄12 Skewness 0 (Excess) kurtosis − 6⁄5 Characteristic function 푒푖푏푡 − 푒푖푎푡

푖푡(푏 − 푎) Other comments Its non-central moments (푟 = 1,2,3, … ) are 퐸(푋푟) = 1 1 (푏푟+1 − 푎푟+1). Its median is (푎 + 푏)⁄2. (푏−1) 푟+1

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “uniform”. For details of other supported probability distributions see here.

The Weibull distribution [WeibullDistribution]

The Weibull distribution is a special case of the generalised extreme value (GEV) distribution. It characterises the distribution of ‘block maxima’ under certain (relatively restrictive) conditions.

Continuous multivariate distributions

The inverse [InverseWishartDistribution]

The inverse Wishart distribution (otherwise called the inverted Wishart distribution) 푊−1(퐕, 푚) is a probability distribution that is used in the Bayesian analysis of real-valued positive definite matrices (e.g. matrices of the type that arise in risk management contexts). It is a conjugate prior for the of a multivariate normal distribution.

It has the following characteristics, where 퐁 is a 푛 × 푛 matrix, 퐕 is a positive and Γ푛(. ) is the multivariate gamma function.

Parameters (and constraints on parameters): 푚 > 푛 − 1 (푚 = degrees of freedom, real) 퐕 > 0 (퐕 = inverse scale matrix, positive definite) Support (i.e. values that it can take) 퐁 > 0, i.e. is positive definite, 퐁 an 푛 × 푛 matrix −1 Probability density function |퐕|푚⁄2|퐁|−(푚+푛+1)⁄2푒−trace(퐕퐁 )⁄2 푚푛⁄2 2 Γ푛(푚⁄2) Mean 퐕

푚 − 푛 − 1

If the elements of 퐁 are 퐵푖,푗 and the elements of 퐕 are 푉푖,푗 then

2 (푚 − 푛 + 1)푉푖,푗 + (푚 − 푛 − 1)푉푖,푖푉푖,푗 var(퐵 ) = 푖,푗 (푚 − 푛)(푚 − 푛 − 1)2(푚 − 푛 − 3)

The main use of the inverse Wishart distribution appears to arise in . Suppose we want to make an inference about a covariance matrix, 퐕, whose prior 푝(퐕) has a 푊−1(퐕, 푚) distribution. If the observation set 퐗 = (퐱1, … , 퐱퐾) where the 퐗푘 are independent 푛-variate Normal (i.e. Gaussian) random variables drawn from a 푁(0, 퐐) distribution then the conditional distribution 푝(퐐|퐗), i.e. the probability of 퐐 given 퐗, has a 푊−1(퐀 + 퐕, 푚 + 퐾) distribution, where 퐀 = 퐗퐗푇 is the sample covariance matrix.

The univariate special case of the inverse Wishart distribution is the inverse gamma distribution. With 푛 = 1, 훼 = 푚⁄2, 훽(= 퐕⁄2) = 푉1,1⁄2, 푥(= 퐁⁄2) = 퐵1,1⁄2 we have:

훽훼푥−훼−1exp(− β⁄x) 푝(푥|훼, 훽) = Γ(α) where Γ(. ) ≡ Γ1(. ) is the ordinary (i.e. univariate) Gamma function, see MnGamma.

For other probability distributions see here.

Copulas (a copula is a special type of continuous multivariate distribution)

The Clayton copula [ClaytonCopula]

The Clayton copula is a copula that allows any specific non-zero level of (lower) tail dependency between individual variables. It is an Archimedean copula, and exchangeable.

Copula name Clayton copula 퐶푙 Common notation 푈~퐶휃 Parameters 휃 ≥ 0 (can be extended to 휃 ≥ −1) Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 Copula −1⁄휃 퐶푙 −휃 퐶휃 (푢1, … , 푢푛) = (∑(푢푖 − 1) + 1) 푖 Or if 휃 = 0 we use the limit lim 퐶퐺퐶 휃→0 휃 Kendall’s rank correlation 휃 coefficient (for bivariate 2 + 휃 case) Coefficient of upper tail 0 dependence, 휆푢 Coefficient of lower tail 2−1⁄휃 dependence, 휆푙 Archimedean generator 휃−1(푡−휃 − 1) function, 휙(푡) Or if 휃 = 0 we use the limit lim 휙휃(푡) = − log 푡. If 휃 ≠ 0 then a 휃→0 simpler version, which does not alter the copula itself, is 휙(푡) = 푡−휃 − 1. Other comments If 휃 = 0 we obtain the independence copula. The Clayton copula (like the Frank copula) is a comprehensive copula in that it interpolates between a lower limit of the countermonotonicity copula (휃 → −1) and an upper limit of the comonotonicity copula (휃 → +∞)..

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Clayton Copula”. For details of other probability distributions see here.

The comonotonicity copula [ComonotonicityCopula]

The Comonotonicity copula is a special copula characterising perfect positive dependence, in the sense that the 푈푖 are almost surely strictly increasing functions of each other.

Copula name comonotonicity copula Common notation 푈~푀 Parameters 휃 Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 Copula 푀(푢1, … , 푢푛) = 푚푖푛(푢1, … 푢푛) Kendall’s rank correlation 1 coefficient (for bivariate case) Coefficient of upper tail 1 dependence Coefficient of lower tail 1 dependence Other comments Is the extreme case where dependency is as strongly “positive” as possible (i.e. achieves the Fréchet upper bound)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Comonotonicity Copula”. For details of other supported probability distributions see here.

The countermonotonicity copula [CountermonotonicityCopula]

The Countermonotonicity copula is a special copula characterising perfect negative dependence, in the sense that 푈1 is almost surely a strictly decreasing function of 푈2 and vice-versa.

Copula name countermonotonicity copula Common notation 푈~푊 Parameters 휃 Domain 0 ≤ 푢푖 ≤ 1 푖 = 1,2 Copula 푊(푢1, 푢2) = max(푢1 + 푢2 − 1,0) Kendall’s rank correlation −1 coefficient Coefficient of upper tail 1 dependence Coefficient of lower tail 1 dependence Other comments Is the extreme case where dependency is as strongly “negative” as possible (i.e. achieves the Fréchet lower bound)

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Countermonotonicity Copula”. For details of other supported probability distributions see here.

The Frank copula [FrankCopula]

The Frank copula is a copula that is sometimes used in the modelling of codependency. It is an Archimedean copula, and exchangeable.

Copula name Frank copula 퐹푟 Common notation 푈~퐶휃 Parameters −∞ < 휃 < ∞ Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 Copula 1 ∏푖(exp(−휃푢푖) − 1) 퐶퐹푟(푢 , … , 푢 ) = − log (1 + ) 휃 1 푛 휃 exp(−휃) − 1 Or if 휃 = 0 we use the limit lim 퐶퐹푟 which the independence copula 휃→0 휃 −1 Kendall’s rank correlation 1 − 4휃 (1 − 퐷1(휃)) coefficient (for bivariate where 퐷1(휃) is the Debye function defined as: case) ∞ −1 푡 퐷1(휃) = 휃 ∫ 푑푡 0 exp(푡) − 1 Coefficient of upper tail 0 dependence, 휆푢 Coefficient of lower tail 0 dependence, 휆푙 Archimedean generator exp(−휃푡) − 1 − log ( ) function, 휙(푡) exp(−휃) − 1 Or if 휃 = 0 we use the limit lim 휙휃(푡) which is taken as − log 푡 휃→0 Other comments If 휃 = 0 we obtain the independence copula. The Frank copula (like the Clayton copula) is a comprehensive copula in that it interpolates between a lower limit of the countermonotonicity copula (휃 → −∞) and an upper limit of the comonotonicity copula (휃 → +∞).

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Frank Copula”. For details of other supported probability distributions see here.

The Generalised Clayton copula [GeneralisedClaytonCopula]

The Generalised Clayton copula is a copula that allows any specific (non-zero) level of (lower) tail dependency between individual variables. It is an Archimedean copula and exchangeable.

Copula name Generalised Clayton copula 퐺퐶 Common notation 푈~퐶휃,훿 Parameters 휃 ≥ 0, 훿 ≥ 1 Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 −1⁄휃 Copula 1⁄훿 퐺퐶 −휃 훿 퐶휃,훿(푢1, … , 푢푛) = ((∑(푢푖 − 1) ) + 1) 푖 Or if 휃 = 0 we use the limit lim 퐶퐺퐶 휃→0 휃 Kendall’s rank correlation (2 + 휃)훿 − 2 coefficient (for bivariate (2 + 휃)훿 case), 휌휏 Coefficient of upper tail 2 − 2−1⁄훿 dependence, 휆푢 Coefficient of lower tail 2−1⁄(휃훿) dependence, 휆푙 Archimedean generator 휃−훿(푡−휃 − 1) function, 휙(푡) Or if 휃 = 0 we use the limit lim 휙휃(푡) 휃→0 Other comments When 훿 = 1 the generalised Clayton copula becomes the Clayton copula.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Generalised Clayton Copula”. For details of other supported probability distributions see here.

The Gumbel copula [GumbelCopula]

The Gumbel copula is a copula that allows any specific level of (upper) tail dependency between individual variables. It is an Archimedean copula, and exchangeable.

Copula name Gumbel copula 퐺푢 Common notation 푈~퐶휃 Parameters 1 ≤ 휃 < ∞ Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 Copula 1⁄휃 퐺푢 휃 퐶휃 (푢1, … , 푢푛) = exp (− (∑(− log 푢푖) ) ) 푖 Kendall’s rank correlation 1 1 − coefficient (for bivariate 휃 case) Coefficient of upper tail 2 − 21⁄휃 dependence, 휆푢 Coefficient of lower tail 0 dependence, 휆푙 Archimedean generator (− log 푡)휃 function, 휙(푡) Other comments If 휃 = 1 we obtain the independence copula and as 휃 → ∞ we approach the comonotonicity copula.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Gumbel Copula”. For details of other supported probability distributions see here.

The Gaussian copula [GaussianCopula]

The Gaussian copula is the copula that underlies the multivariate normal distribution.

Copula name Gaussian copula 퐺푎 Common notation 푈~퐶퐶 Parameters 퐶, a non-negative definite 푛 × 푛 matrix, i.e. a matrix that can correspond to a correlation matrix Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 퐺푎 −1 −1 Copula 퐶퐶 (푢1, … , 푢푛) = 횽퐶(푁 (푢1), … , 푁 (푢2)) −1 where 푁 (푥) is the inverse normal function and 횽퐶 is the cumulative distribution function of the multivariate normal distribution defined by a covariance matrix equal to 퐶 Kendall’s rank correlation 2 arcsin 휌 coefficient (for bivariate 휋 case), 휌휏 Where 휌 is the correlation coefficient between the two variables Coefficient of upper tail 0 (unless the correlation matrix exhibits perfect positive or negative dependence, 휆푢 dependence) Coefficient of lower tail 0 (unless the correlation matrix exhibits perfect positive or negative dependence, 휆푙 dependence) Other comments The Spearman rank correlation coefficient is given by: 6 휌 휌 = arcsin 푆 휋 2 where 휌 is the (normal) correlation coefficient between the two variables.

If 퐶 = 퐼푛 (the 푛 × 푛 identity matrix) then we obtain the independence copula.

Nematrian web functions

Functions relating to the above distribution in the two dimensional case may be accessed via the Nematrian web function library by using a DistributionName of “Gaussian Copula (2d)”. For details of other supported probability distributions see here.

The Independence copula [IndependenceCopula]

The Independence copula is the copula that results from a dependency structure in which each individual variable is independent of each other. It is an Archimedean copula, and exchangeable.

Copula name Independence copula Common notation 푈~Π Parameters None Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 Copula Π(푢1, … , 푢푛) = ∏ 푢푖 푖 Kendall’s rank correlation 0 coefficient (for bivariate case) Coefficient of upper tail 0 dependence, 휆푢 Coefficient of lower tail 0 dependence, 휆푙 Archimedean generator − log 푡 function, 휙(푡) Other comments The independence copula is a special case of several Archimedean copulas. It is also the special case of the Gaussian copula with a correlation matrix equal to the identity matrix.

Nematrian web functions

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “Independence Copula”. For details of other supported probability distributions see here.

The t copula [TCopula]

The t copula is the copula that underlies the multivariate Student’s t distribution.

Copula name t copula 푡 Common notation 푈~퐶휐,퐶 Parameters 퐶, a non-negative definite 푛 × 푛 matrix, i.e. a matrix that can correspond to a correlation matrix 휈 = degrees of freedom (휈 > 0, usually 휈 is an integer although in some situations a non-integral 휈 can arise) (note in principle each could in principle have a different number of degrees of freedom although such a refinement is not commonly seen) Domain 0 ≤ 푢푖 ≤ 1 푖 = 1, … , 푛 푡 −1 −1 Copula 퐶휐,퐶(푢1, … , 푢푛) = 풕휐,퐶(푡휐 (푢1), … , 푡휐 (푢2)) −1 where 푡휐 (푥) is the inverse student’s t function and 풕휐,퐶 is the cumulative distribution function of the multivariate student’s t distribution with arbitrary mean and matrix generator equal to 퐶 Kendall’s rank correlation 2 arcsin 휌 coefficient (for bivariate 휋 case), 휌휏 Where 휌 is the correlation coefficient between the two variables Coefficient of upper tail (휐 + 1)(1 − 휌) dependence, 휆푢 2푡 (−√ ) 휐+1 1 + 휌 Coefficient of lower tail (휐 + 1)(1 − 휌) dependence, 휆푙 2푡 (−√ ) 휐+1 1 + 휌

Other comments If 퐶 = 퐼푛 (the 푛 × 푛 identity matrix) then, in contrast to the Gaussian copula, we do not recover the independence copula.

Nematrian web functions

Functions relating to the above distribution in the two-dimensional case may be accessed via the Nematrian web function library by using a DistributionName of “student’s t (2d)”. For details of other supported probability distributions see here.